Question: Simplify the following expression: $ r = \dfrac{4z}{6z + 5} + \dfrac{-8}{9} $
Solution: In order to add expressions, they must have a common denominator. Multiply the first expression by $\dfrac{9}{9}$ $ \dfrac{4z}{6z + 5} \times \dfrac{9}{9} = \dfrac{36z}{54z + 45} $ Multiply the second expression by $\dfrac{6z + 5}{6z + 5}$ $ \dfrac{-8}{9} \times \dfrac{6z + 5}{6z + 5} = \dfrac{-48z - 40}{54z + 45} $ Therefore $ r = \dfrac{36z}{54z + 45} + \dfrac{-48z - 40}{54z + 45} $ Now the expressions have the same denominator we can simply add the numerators: $r = \dfrac{36z - 48z - 40}{54z + 45} $ $r = \dfrac{-12z - 40}{54z + 45}$